Lemma 5 (Lemma 6, Corollary 7 and Corollary 8 in [1]} combined) 

Suppose \(({\mathbf {x}}_1,\dots ,{\mathbf {x}}_n) \in \mathbb {R}^d\)  satisfy condition:nSG. Then, there exists an absolute constant \(c\)  such that for any fixed \(\delta \in (0,1)\) , \(\theta > 0\) , with probability at least \(1 - \delta \) ,
\(*{\sum _{i=1}^n {\mathbf {x}}_i} \le c \theta \sum _{i=1}^n \sigma _i^2 + \frac{1}{\theta } \log (2d/\delta ).\) 
